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This website is specially built to complement the learning of the '''Mechanics''' course in the bachelor's degrees of '''[https://etseib.upc.edu/en Barcelona School of Industrial Engineering (ETSEIB)]''' of the '''[https://www.upc.edu/en Polytechnic University of Catalonia (UPC) · BarcelonaTech]'''. | |||
== | This aims to be an '''accessible''' and '''interactive''' '''open tool'''. It's development started on 2022 and it gathers more than 50 years of teaching experience. It's content is organized in brief units which contain the fundamental concepts and some fully worked-out examples. Some simple mathematical proofs are included, but the longer or complex ones are refered to biblography. | ||
The contentent is focused on '''general space movement of rigid bodies and muli-body systems''', but '''particles''' are also considered. Dynamics formulation is vectorial, due to the relevance of the force vector in mechanical engineering. The last units are an introduction to energetics. | |||
<small> | |||
'''About the status of the site''': | |||
This project has been developed with limited resources, both technical and human. Nowadays, the server presents some issues, so in case any error may appear, we kindly invite the users to refresh the page and continue enjoying the content 😊. | |||
The best experience will be through a computer or a tablet 📵. | |||
The site is still under construction and some interactive resources and videos are still to be uploaded. Also, the Dynamics and Energetics blocks are still to be published. Having said that, it already is a good tool to help in the process of learning Mechanics 🎯. | |||
</small> | |||
<p align="right"><small>© Universitat Politècnica de Catalunya. [[Mechanics:Copyrights |All rights reserved]]</small></p> | |||
__NOTOC__ | |||
------------ | |||
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===[[Introduction]]=== | |||
:: [[Introduction#I.1 What is mechanics?|I.1 What is mechanics?]] | |||
:: [[Introduction#I.2 Models for material objects|I.2 Models for material objects]] | |||
:: [[Introduction#I.3 Limitations of Newtonian mechanics|I.3 Limitations of Newtonian mechanics]] | |||
:: [[Introduction#I.4 Reference frame|I.4 Reference frame]] | |||
===[[Vector calculus]]=== | |||
::[[Vector calculus#V.1 Geometric representation of a vector|V.1 Geometric representation of a vector]] | |||
::[[Vector calculus#V.2 Operations between vectors with geometric representation|V.2 Operations between vectors with geometric representation]] | |||
::::<small>[[Vector calculus#Instantaneous operations: addition, scalar product, vector product|Instantaneous operations: addition, scalar product, vector product]]</small> | |||
::::<small>[[Vector calculus#Operations along time: time derivative|Operations along time: time derivative]]</small> | |||
::[[Vector calculus#V.3 Analytical representation of a vector|V.3 Analytical representation of a vector]] | |||
::[[Vector calculus#V.4 Operations between vectors with analytical representation|V.4 Operations between vectors with analytical representation]] | |||
::::<small>[[Vector calculus#Instantaneous operations: addition, scalar product, vector product|Instantaneous operations: addition, scalar product, vector product]]</small> | |||
::::<small>[[Vector calculus#Operations along time: time derivative|Operations along time: time derivative]]</small> | |||
==KINEMATICS== | |||
===[[C1. Configuration of a mechanical system]]=== | |||
:: [[C1. Configuration of a mechanical system#C1.1 Position of a particle|C1.1 Position of a particle]] | |||
:: [[C1. Configuration of a mechanical system#C1.2 Configuration of a rigid body|C1.2 Configuration of a rigid body]] | |||
:: [[C1. Configuration of a mechanical system#C1.3 Orientation of a rigid body with planar motion|C1.3 Orientation of a rigid body with planar motion]] | |||
:: [[C1. Configuration of a mechanical system#C1.4 Orientation of a rigid body moving in space|C1.4 Orientation of a rigid body moving in space]] | |||
::::<small>[[C1. Configuration of a mechanical system#Rotations about fixed axes|Rotations about fixed axes]]</small> | |||
::::<small>[[C1. Configuration of a mechanical system#Euler rotations|Euler rotations]]</small> | |||
:: [[C1. Configuration of a mechanical system#C1.5 Independent coordinates|C1.5 Independent coordinates]] | |||
===[[C2. Movement of a mechanical system]]=== | |||
:: [[C2. Movement of a mechanical system#C2.1 Velocity of a particle|C2.1 Velocity of a particle]] | |||
:: [[C2. Movement of a mechanical system#C2.2 Acceleration of a particle|C2.2 Acceleration of a particle]] | |||
:: [[C2. Movement of a mechanical system#C2.3 Intrinsic components of the acceleration|C2.3 Intrinsic components of the acceleration]] | |||
:: [[C2. Movement of a mechanical system#C2.4 Angular velocity of a rigid body|C2.4 Angular velocity of a rigid body]] | |||
::::<small>[[C2. Movement of a mechanical system#Simple rotation|Simple rotation]]</small> | |||
::::<small>[[C2. Movement of a mechanical system#Rotation in space|Rotation in space (Rotacions d'Euler)]]</small> | |||
:: [[C2. Movement of a mechanical system#C2.5 Angular acceleration of a rigid body|C2.5 Angular acceleration of a rigid body]] | |||
:: [[C2. Movement of a mechanical system#C2.6 Particle kinematics versus rigid body kinematicsrígid|C2.6 Particle kinematics versus rigid body kinematics]] | |||
:: [[C2. Movement of a mechanical system#C2.7 Degrees of freedom of a mechanical system|C2.7 Degrees of freedom of a mechanical system]] | |||
:: [[C2. Movement of a mechanical system#C2.8 Usual constraints in mechanical systems|C2.8 Usual constraints in mechanical systems]] | |||
:: [[C2. Movement of a mechanical system#C2.E General examples|C2.E General examples]] | |||
===[[C3. Composition of movements]]=== | |||
:: [[C3. Composition of movements#C3.1 Composition of velocities|C3.1 Composition of velocities]] | |||
:: [[C3. Composition of movements#C3.2 Composition of accelerations|C3.2 Composition of accelerations]] | |||
:: [[C3. Composition of movements#C3.3 Composition versus time derivative|C3.3 Composition versus time derivative]] | |||
:: [[C3. Composition of movements#C3.E General examples|C3.E General examples]] | |||
===[[C4. Rigid body kinematics]]=== | |||
:: [[C4. Rigid body kinematics#C4.1 Velocity distribution|C4.1 Velocity distribution]] | |||
:: [[C4. Rigid body kinematics#C4.2 Accelerations distribution|C4.2 Accelerations distribution]] | |||
:: [[C4. Rigid body kinematics#C4.3 Geometry of the velocity distribution: Instantaneous Screw Axis (ISA)|C4.3 Geometry of the velocity distribution: Instantaneous Screw Axis (ISA)]] | |||
::[[C4. Rigid body kinematics#C4.4 Fixed axode and moving axode|C4.4 Fixed axode and moving axode]] | |||
::[[C4. Rigid body kinematics#C4.E General examples|C4.E General examples]] | |||
===[[C5. Rigid body kinematics: planar motion]]=== | |||
:: [[C5. Rigid body kinematics: planar motion#C5.1 Instantaneous Center of Rotation (ICR)|C5.1 Instantaneous Center of Rotation (ICR)]] | |||
:: [[C5. Rigid body kinematics: planar motion#C5.2 Examples|C5.2 Examples]] | |||
:: [[C5. Rigid body kinematics: planar motion#C5.3 Introduction to vehicle kinematics|C5.3 Introduction to vehicle kinematics]] | |||
:: [[C5. Rigid body kinematics: planar motion#C5.E General examples|C5.E General examples]] | |||
==DYNAMICS== | |||
===[[D1. Foundational laws of Newtonian dynamics#|D1. Foundational laws of Newtonian dynamics]]=== | |||
::[[D1. Foundational laws of Newtonian dynamics#D1.1 Galilean reference frames|D1.1 Galilean reference frames]] | |||
::[[D1. Foundational laws of Newtonian dynamics#D1.2 Galileo’s Principle of Relativity|D1.2 Galileo’s Principle of Relativity]] | |||
::[[D1. Foundational laws of Newtonian dynamics#D1.3 Newton’s Principle of Determinacy|D1.3 Newton’s Principle of Determinacy]] | |||
::[[D1. Foundational laws of Newtonian dynamics#D1.4 Newton’s first law (inertia law)|D1.4 Newton’s first law (inertia law)]] | |||
::[[D1. Foundational laws of Newtonian dynamics#D1.5 Newton’s second law (fundamental law of dynamics)|D1.5 Newton’s second law (fundamental law of dynamics)]] | |||
::[[D1. Foundational laws of Newtonian dynamics#D1.6 Newton’s third law (action-reaction principle)|D1.6 Newton’s third law (action-reaction principle)]] | |||
::[[D1. Foundational laws of Newtonian dynamics#D1.7 Particle dynamics in non Galilean reference frames|D1.7 Particle dynamics in non Galilean reference frames]] | |||
===[[D2. Interaction forces between particles#|D2. Interaction forces between particles]]=== | |||
::[[D2. Interaction forces between particles#D2.1 Kinematic dependence of interaction forces|D2.1 Kinematic dependence of interaction forces]] | |||
::[[D2. Interaction forces between particles#D2.2 Classification of interaction forces|D2.2 Classification of interaction forces]] | |||
::[[D2. Interaction forces between particles#D2.3 Gravitational attraction|D2.3 Gravitational attraction]] | |||
::[[D2. Interaction forces between particles#D2.4 Interaction through springs|D2.4 Interaction through springs]] | |||
::[[D2. Interaction forces between particles#D2.5 Interaction through dampers|D2.5 Interaction through dampers]] | |||
::[[D2. Interaction forces between particles#D2.6 Interaction through actuators|D2.6 Interaction through actuators]] | |||
::[[D2. Interaction forces between particles#D2.7 Constraint interactions|D2.7 Constraint interactions]] | |||
::[[D2. Interaction forces between particles#D2.8 Friction|D2.8 Friction]] | |||
===[[D3. Interactions between rigid bodies#|D3. Interactions between rigid bodies]]=== | |||
::[[D3. Interactions between rigid bodies#D3.1 Torsor associated with a system of forces|D3.1 Torsor associated with a system of forces]] | |||
::[[D3. Interactions between rigid bodies#D3.2 Gravitational attraction|D3.2 Gravitational attraction]] | |||
::[[D3. Interactions between rigid bodies#D3.3 Interaction through linear and torsion springs and dampers|D3.3 Interaction through linear and torsion springs and dampers]] | |||
::[[D3. Interactions between rigid bodies#D3.4 Direct constraint interactions|D3.4 Direct constraint interactions]] | |||
::[[D3. Interactions between rigid bodies#D3.5 Indirect constraint interactions: Constraint Auxiliary Elements (CAE)|D3.5 Indirect constraint interactions: Constraint Auxiliary Elements (CAE)]] | |||
::[[D3. Interactions between rigid bodies#D3.6 Interaccions per mitjà d’actuadors lineals i rotacionals|D3.6 Interaction through linear and rotatory actuators]] | |||
===[[D4. Vector theorems#|D4. Vector theorems]]=== | |||
::[[D4. Vector theorems#D4.1 Linear Momentum Theorem (LMT) in Galilean reference frames|D4.1 Linear Momentum Theorem (LMT) in Galilean reference frames]] | |||
::[[D4. Vector theorems#D4.2 LMT: application examples|D4.2 LMT: application examples]] | |||
::[[D4. Vector theorems#D4.3 Linear Momentum Theorem (LMT) in non Galilean reference frames|D4.3 Linear Momentum Theorem (LMT) in non Galilean reference frames]] | |||
::[[D4. Vector theorems#D4.4 Angular Momentum Theorem (AMT): general formulation|D4.4 Angular Momentum Theorem (AMT): general formulation]] | |||
::[[D4. Vector theorems#D4.5 Angular Momentum Theorem (AMT): particular formulations|D4.5 Angular Momentum Theorem (AMT): particular formulations]] | |||
::[[D4. Vector theorems#D4.6 AMT: application examples|D4.6 AMT: application examples]] | |||
::[[D4. Vector theorems#D4.7 Dynamics of Constraint Auxiliary Elements|D4.7 Dynamics of Constraint Auxiliary Elements]] | |||
::[[D4. Vector theorems#D4.8 Barycentric decomposition of the angular momentum|D4.8 Barycentric decomposition of the angular momentum]] | |||
===[[D5. Mass distribution#|D5. Mass distribution]]=== | |||
::[[D5. Mass distribution#D5.1 Centre of masses|D5.1 Centre of masses]] | |||
::[[D5. Mass distribution#D5.2 Inertia tensor|D5.2 Inertia tensor]] | |||
::[[D5. Mass distribution#D5.3 Some relevant properties of the inertia tensor|D5.3 Some relevant properties of the inertia tensor]] | |||
::[[D5. Mass distribution#D5.4 Steiner’s Theorem|D5.4 Steiner’s Theorem]] | |||
::[[D5. Mass distribution#D5.5 Change of vector basis|D5.5 Change of vector basis]] | |||
===[[D6. Examples of 2D dynamics#|D6. Examples of 2D dynamics]]=== | |||
::[[D6. Examples of 2D dynamics#D6.1 2D kinematics and 2D dynamics|D6.12D kinematics and 2D dynamics]] | |||
::[[D6. Examples of 2D dynamics#D6.2 Free-body diagram (FBD) and roadmap|D6.2 Free-body diagram (FBD) and roadmap]] | |||
::[[D6. Examples of 2D dynamics#D6.3 Examples with just one rigid body|D6.3 Examples with just one rigid body]] | |||
::[[D6. Examples of 2D dynamics#D6.4 General diagram of interactions (GDI)|D6.4 General diagram of interactions (GDI)]] | |||
::[[D6. Examples of 2D dynamics#D6.5 Examples of multibody systems|D6.5 Examples of multibody systems]] | |||
===[[D7. Examples of 3D dynamics#|D7. Examples of 3D dynamics]]=== | |||
::[[D7. Examples of 3D dynamics#D7.1 Analysis of the equations of motion|D7.1 Analysis of the equations of motion]] | |||
::[[D7. Examples of 3D dynamics#D7.2 General examples|D7.2 General examples]] | |||
===[[D8. Conservation of dynamic magnitudes#|D8. Conservation of dynamic magnitudes]]=== | |||
::[[D8. Conservation of dynamic magnitudes#D8.1 Examples|D8.1 Examples]] | |||
==ENERGETICS== | |||
===[[E1. Work-Energy Theorem: differential form#|E1. Work-Energy Theorem: differential form]]=== | |||
::[[E1. Work-Energy Theorem: differential form#E1.1 Power balance in a system of particles|E1.1 Power balance in a system of particles]] | |||
::[[E1. Work-Energy Theorem: differential form#E1.2 Power of an action-reaction pair|E1.2 Power of an action-reaction pair]] | |||
::[[E1. Work-Energy Theorem: differential form#E1.3 Power of a system of forces on a rigid body|E1.3 Power of a system of forces on a rigid body]] | |||
::[[E1. Work-Energy Theorem: differential form#E1.4 Power balance in a multibody system: direct and indirect calculation|E1.4 Power balance in a multibody system: direct and indirect calculation]] | |||
----------- | |||
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==Authors== | |||
<center><gallery> | |||
File:Autors Ana.png|alt=Ana Barjau Condomines|Ana Barjau Condomines|link=https://directori.upc.edu/directori/dadesPersona.jsp?id=1001000 | |||
File:Lluis Ros2.png|alt=Lluís Ros Giralt|Lluís Ros Giralt|link=https://directori.upc.edu/dadesPersona/1002527 | |||
File:Autors Ernest4.jpg|alt=Ernest Bosch Soldevila|Ernest Bosch Soldevila|link=https://www.linkedin.com/in/ernest-bosch-soldevila/ | |||
</gallery></center> | |||
::'''Ilustrations:''' | |||
<center><gallery> | |||
File:Autors_Joaquim.png|alt=Joaquim Agulló i Batlle|Joquim Agulló i Batlle|link=https://www.agullobatlle.cat/ | |||
</gallery></center> | |||
::'''Collaborators:''' | |||
<center><gallery> | |||
File:Autors_Daniel.png|alt=Daniel Clos Costa|Daniel Clos Costa|link=https://directori.upc.edu/directori/dadesPersona.jsp?id=1002252 | |||
File:Autors_Rosa.png|alt=Rosa Pàmies Vilà|Rosa Pàmies Vilà|link=https://directori.upc.edu/directori/dadesPersona.jsp?id=1066910 | |||
File:Autors_Albert.png|alt=Albert Peiret Giménez|Albert Peiret Giménez|link=https://directori.upc.edu/directori/dadesPersona.jsp?id=1115007 | |||
File:Autors_Javier.png|alt=Javier Sistiaga Vidal-Ribas|Javier Sistiaga Vidal-Ribas|link=https://directori.upc.edu/directori/dadesPersona.jsp?id=1114855 | |||
</gallery></center> | |||
::'''Editing and interactive animations:''' | |||
<center><gallery> | |||
File:Autors_Arnau.png|alt=Arnau Marzábal Gatell|Arnau Marzábal Gatell|link=https://www.linkedin.com/in/arnau-marzabal/ | |||
File:Autors_Berta.png|alt=Berta Ros Blanco|Berta Ros Blanco|link=https://www.linkedin.com/in/berta-ros/ | |||
File:Anna Pons.png|alt=Ana Pons Ferrer|Ana Pons Ferrer|link=https://www.linkedin.com/in/anaponsferrer/ | |||
File:Jana.jpg|alt=Jana Bonet|Jana Bonet Fernández|link=https://www.linkedin.com/in/jana-bonet-fern%C3%A1ndez-980aa51b9 | |||
File:Gemma3.jpg|alt=Gemma Izquierdo|Gemma Izquierdo Santamaria|link=https://es.linkedin.com/in/gemma-izquierdo-santamaria-b5a8ba240 | |||
</gallery></center> | |||
<center> | |||
[https://www.youtube.com/channel/UCqWvnHTViRPI1wHlUQXqH-Q '''Mechanics Lab'''] ''' - ''' [https://etseib.upc.edu/en '''Barcelona School of Industrial Engineering (ETSEIB)'''] | |||
[https://em.upc.edu/en '''Mechanical Engineering Department'''] ''' - ''' [https://www.upc.edu/en '''Polytechic University of Catalonia (UPC) · BarcelonaTech'''] | |||
</center> | |||
------------ | |||
------------ | |||
==Bibliographic references== | |||
Batlle, J. A., Barjau, A. (2020) “'''Rigid Body Kinematics'''” Cambridge Univerity Press. ISBN: 978-1-108-47907-3 | |||
Batlle, J. A., Barjau, A. (2022) “'''Rigid Body Dynamics'''” Cambridge Univerity Press. ISBN: 978-1-108-84213-6 | |||
Agulló, J. (2002) “'''Mecànica de la partícula i del sòlid rígid'''" Publicacions OK Punt. ISBN: 84-920850-6-1 (''Disponible en accés obert al [https://www.agullobatlle.cat/activitat-docent '''web de l'autor''']) | |||
Agulló, J. (2000) “'''Mecánica de la partícula i del sólido rígido'''" Publicacions OK Punt. ISBN: 84-920850-5-3 (''Disponible en accés obert al [https://www.agullobatlle.cat/activitat-docent '''web de l'autor''']) | |||
<center> | |||
<gallery> | |||
File:RBK portada.png|alt=Rigid body kinematics|link=https://www.cambridge.org/es/academic/subjects/engineering/engineering-design-kinematics-and-robotics/rigid-body-kinematics?format=HB&isbn=9781108479073 | |||
File:RBD portada.png|alt=Rigid body dynamics|link=https://www.cambridge.org/es/academic/subjects/engineering/engineering-design-kinematics-and-robotics/rigid-body-dynamics?format=HB&isbn=9781108842136 | |||
File:Llibre verd.png|alt=Mecànica de la partícula i del sòlid rígid|link=https://www.agullobatlle.cat/activitat-docent | |||
File:Llibre vermell.jpg|alt=Mecánica de la partícula i del sólido rígido|link=https://www.agullobatlle.cat/activitat-docent | |||
</gallery> | |||
</center> | |||
--------- | |||
--------- | |||
[[File:Logo Lab Mec horitzontal.png|thumb|center|500px|link=https://em.upc.edu/en| ]] | |||
Latest revision as of 07:09, 13 May 2025
This website is specially built to complement the learning of the Mechanics course in the bachelor's degrees of Barcelona School of Industrial Engineering (ETSEIB) of the Polytechnic University of Catalonia (UPC) · BarcelonaTech.
This aims to be an accessible and interactive open tool. It's development started on 2022 and it gathers more than 50 years of teaching experience. It's content is organized in brief units which contain the fundamental concepts and some fully worked-out examples. Some simple mathematical proofs are included, but the longer or complex ones are refered to biblography.
The contentent is focused on general space movement of rigid bodies and muli-body systems, but particles are also considered. Dynamics formulation is vectorial, due to the relevance of the force vector in mechanical engineering. The last units are an introduction to energetics.
About the status of the site:
This project has been developed with limited resources, both technical and human. Nowadays, the server presents some issues, so in case any error may appear, we kindly invite the users to refresh the page and continue enjoying the content 😊.
The best experience will be through a computer or a tablet 📵.
The site is still under construction and some interactive resources and videos are still to be uploaded. Also, the Dynamics and Energetics blocks are still to be published. Having said that, it already is a good tool to help in the process of learning Mechanics 🎯.
© Universitat Politècnica de Catalunya. All rights reserved
Introduction
Vector calculus
KINEMATICS
C1. Configuration of a mechanical system
C2. Movement of a mechanical system
- C2.1 Velocity of a particle
- C2.2 Acceleration of a particle
- C2.3 Intrinsic components of the acceleration
- C2.4 Angular velocity of a rigid body
- C2.5 Angular acceleration of a rigid body
- C2.6 Particle kinematics versus rigid body kinematics
- C2.7 Degrees of freedom of a mechanical system
- C2.8 Usual constraints in mechanical systems
- C2.E General examples
C3. Composition of movements
C4. Rigid body kinematics
C5. Rigid body kinematics: planar motion
DYNAMICS
D1. Foundational laws of Newtonian dynamics
- D1.1 Galilean reference frames
- D1.2 Galileo’s Principle of Relativity
- D1.3 Newton’s Principle of Determinacy
- D1.4 Newton’s first law (inertia law)
- D1.5 Newton’s second law (fundamental law of dynamics)
- D1.6 Newton’s third law (action-reaction principle)
- D1.7 Particle dynamics in non Galilean reference frames
D2. Interaction forces between particles
D3. Interactions between rigid bodies
- D3.1 Torsor associated with a system of forces
- D3.2 Gravitational attraction
- D3.3 Interaction through linear and torsion springs and dampers
- D3.4 Direct constraint interactions
- D3.5 Indirect constraint interactions: Constraint Auxiliary Elements (CAE)
- D3.6 Interaction through linear and rotatory actuators
D4. Vector theorems
- D4.1 Linear Momentum Theorem (LMT) in Galilean reference frames
- D4.2 LMT: application examples
- D4.3 Linear Momentum Theorem (LMT) in non Galilean reference frames
- D4.4 Angular Momentum Theorem (AMT): general formulation
- D4.5 Angular Momentum Theorem (AMT): particular formulations
- D4.6 AMT: application examples
- D4.7 Dynamics of Constraint Auxiliary Elements
- D4.8 Barycentric decomposition of the angular momentum
D5. Mass distribution
D6. Examples of 2D dynamics
D7. Examples of 3D dynamics
D8. Conservation of dynamic magnitudes
ENERGETICS
E1. Work-Energy Theorem: differential form
Authors
Ana Barjau Condomines
Lluís Ros Giralt
Ernest Bosch Soldevila
- Ilustrations:
Joquim Agulló i Batlle
- Collaborators:
Daniel Clos Costa
Rosa Pàmies Vilà
Albert Peiret Giménez
Javier Sistiaga Vidal-Ribas
- Editing and interactive animations:
Arnau Marzábal Gatell
Berta Ros Blanco
Ana Pons Ferrer
Jana Bonet Fernández
Gemma Izquierdo Santamaria
Mechanics Lab - Barcelona School of Industrial Engineering (ETSEIB)
Mechanical Engineering Department - Polytechic University of Catalonia (UPC) · BarcelonaTech
Bibliographic references
Batlle, J. A., Barjau, A. (2020) “Rigid Body Kinematics” Cambridge Univerity Press. ISBN: 978-1-108-47907-3
Batlle, J. A., Barjau, A. (2022) “Rigid Body Dynamics” Cambridge Univerity Press. ISBN: 978-1-108-84213-6
Agulló, J. (2002) “Mecànica de la partícula i del sòlid rígid" Publicacions OK Punt. ISBN: 84-920850-6-1 (Disponible en accés obert al web de l'autor)
Agulló, J. (2000) “Mecánica de la partícula i del sólido rígido" Publicacions OK Punt. ISBN: 84-920850-5-3 (Disponible en accés obert al web de l'autor)
